Given Theorem REF  and as already observed by Tracy–Widom and Borodin–Okounkov [1]}, [2]} in the context of Toeplitz determinants and the classical strong Szegő limit theorem, the limits are consequences of the Cauchy identities
\(\sum _{\lambda } sp_{\lambda } (\rho ^+) s_{\lambda } (\rho ^-) = Z_{sp}, \qquad \sum _{\lambda } o_{\lambda } (\rho ^+) s_{\lambda } (\rho ^-) = Z_{o}.\) 
